Z-score calculator
A z-score counts how many standard deviations a value sits from the mean. Type any one of the three and get the other two.
- z = (x − μ) / σ = (650 − 500) / 100 = 1.5
- Area below z = Φ(1.5) = 0.9332, so 650 is higher than about 93.3% of values.
What a z-score tells you
A z-score re-measures a value in standard deviations from the mean. That turns scores on different scales into one common scale, which is how you can compare an SAT section score with an ACT score, or one child's height with the growth charts.
z = (x − μ) / σ and back again: x = μ + z·σ
Worked example: which result is better?
Maya scores 650 on a test with mean 500 and SD 100, so z = 1.5. Leo scores 28 on a test with mean 21 and SD 5, so z = 1.4. Maya's result is slightly more exceptional relative to her test-takers, at about the 93rd percentile versus Leo's 92nd.
Rules of thumb
- |z| < 1: ordinary, like about two-thirds of values.
- |z| between 2 and 3: unusual, in the outer 5%.
- |z| > 3: very rare for normal data; check for an error or a genuinely special case.
The percentile conversion assumes the data are roughly normal. For real data of any shape, use the percentile calculator on the raw numbers instead.
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Questions students ask
What is the z-score formula?
z = (x − μ) / σ, where x is the value, μ the mean and σ the standard deviation. For a sample, use the sample mean x̄ and sample standard deviation s.
What is a good z-score?
It depends on the direction you care about. z = 0 is exactly average; z = 1 beats about 84% of a normal population; z = 2 beats about 97.7%. Values beyond ±3 are unusual.
Can a z-score be negative?
Yes. A negative z-score just means the value is below the mean. z = −1.5 is one and a half standard deviations below average, at roughly the 6.7th percentile.
How do I turn a z-score into a percentile?
Find the cumulative normal area to the left of z (the z table cell) and multiply by 100. z = 0.84 gives 0.7995, about the 80th percentile.